# nLab hyperbolic manifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

A hyperbolic space is the analog of a Euclidean space as one passes from Euclidean geometry to hyperbolic geometry. The generalization of the concept of hyperbolic plane to higher dimension.

A hyperbolic manifold is a Riemannian manifold $(X,g)$ of constant sectional curvature $-1$.

## Properties

### Zeta functions

There are canonical zeta functions associated with suitable (odd-dimensional) hyperbolic manifolds, see at Selberg zeta function and Ruelle zeta function.

### Relation to Chern-Simons theory

There is a curious relation of volumes of hyperbolic 3-manifolds to the action functional of Chern-Simons theory/Dijkgraaf-Witten theory.

Let $G$ be a Lie group and $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$ a cocycle in degree-3 (generalized) Lie group cohomology. Write $\flat G$ for the underlying discrete group and $\flat \mathbf{c} \colon \mathbf{B} \flat G \to \mathbf{B}^3 \flat U(1)$ for the induced cocycle in ordinary (discrete) group cohomology, $[\flat \mathbf{c}] \in H^3_{Grp}(G_{disc},U(1)_{disc})$.

Then for $\Sigma$ a closed manifold of dimension 3, a map (of smooth infinity-groupoids) $\Sigma \to \mathbf{B}\flat G$ is a flat $G$-principal connection on $\Sigma$ and the composite

$[\Sigma, \mathbf{B}\flat G] \stackrel{[\Sigma, \flat \mathbf{c}]}{\to} [\Sigma, \mathbf{B}^3 \flat U(1)] \stackrel{\int_{\Sigma}}{\to} U(1)$

is the action functional for $G$-Chern-Simons theory on $\Sigma$ restricted to $G$-flat connections, or equivalently is the action functional of $\flat G$-Dijkgraaf-Witten theory.

Now for $G = SL(n,\mathbb{C})$ the complex special linear group and hence for Chern-Simons theory with complex gauge group, it turns out that the imaginary part of this flat Chern-Simons/Dijkgraaf-Witten invariant of 3-manifolds always has an expression as a combination of volumes of hyperbolic 3-manifolds.

For $n = 2$ this is well understood conceptually. For $n \geq 3$ it has been checked by the computer algebra? system SnapPy (Zickert 07), but the conceptual reason remains unclear.

The combination $CS(\omega) + i vol$ is also called the complex volume or Cheeger-Simons class of a hyperbolic 3-manifold (e.g. Neumann 11, section 2.3. Garoufalidis-Thurston-Zickert 11).

(It is maybe noteworthy that the same kind of combination appears as the contribution of membrane instantons.)

### Mostow rigidity theorem

The Mostow rigidity theorem states that every hyperbolic manifold of dimension $\geq 3$ and of finite volume is uniquely determined by its fundamental group.

A Riemannian manifold

• with zero sectional curvature is a Euclidean manifold?;

• with +1 sectional curvature is an elliptic manifold?